[ex.] 1. I have no allergies or any medical issues.
2. John serves a chicken with no sauce or any kind of seasoning.

I suspect that such a use is wrong from a prescriptive grammatical point of view, because "no" modifies "allergies" in (ex.1) and the sentence is split as follows: "I have no allergies" or "I have any medical issues". But this is not what sentence (ex.1) means.
(ex.1) means that I have no allergies or (and?) no medical issues.

Probably, I think that (ex.1) is reanalyzed as "I don't have any allergies or any medical issues".

My question is:
(1) Is the use like (ex.) correct (in prescriptive grammatical terms)?
(2) If not, what is the correct expression to tell the expressions like (ex.1) and (ex.2)?
(3) Or is there some grammatical rule where when "no" is used, the word that comes after "or" may be "any".
(4) Given that I am wrong and the use like (ex.) is correct, why is this so?

It's a quantifier (which is what some is, too), and has interesting interactions with negatives -- the example sentence is an example of one formulation of DeMorgan's Law: Not (p Or q) ≣ (Not p) And (Not q). Another formulation is Not (For Some/Any p) ≣ (For All p) (Not p). See umich.edu/~jlawler/logicguide.pdf for more.
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John LawlerMay 10 '12 at 16:50

Note that in both these examples the word "other" is at least implicit (even if not actually present), in that both "denied possibilities" (allergies|medical issues, and sauce|kind of seasoning) consist of a specific example, followed by the general classification for "other things of that same type".

The word "any" simply conveys optional emphasis (definitely nothing of the type referred to applies). I feel this strengthens the case for using nor rather than or, because nor also emphasises the negatory nature of the statement.

It's a fine point, but looking at a similar construction without an implicit "other"...

I have no ability or|nor [any] desire to answer this question...

...it seems to me that in the absence of any other context, we might more often expect or to be used if the statement continued with something like "...but since I'm legally obliged to, I will answer". Whereas the more explicitly negative nor might be followed by "...and therefore refuse to answer".

The construct no NOUN or any NOUN is commonly used with the same general meaning as no NOUN or NOUN. It is parsed according to logical rules: no (A or B) => (no A) and (no B).

I have no chicken or fish.

I have no wealth or ambition.

While I was instructed in grade school (as a native U.S. speaker) that I could say I have neither chicken nor fish I do not recall ever being told not to say I have no chicken or fish. If it had been at some point considered informal or wrong to say I have no chicken or fish it does not matter today as it is by far the most common usage.

Adding any to the beginning of the second noun phrase is generally short for any other and is used to prevent confusion as to why you mentioned two things where the first is part of the second. If I say I have no chicken or food it seems like I'm implying chicken is not food, so I'll say I have no chicken or any (other) food.

In your first example, "allergies" could be considered a medical issue, so people hedge by including any. (Indeed, the reason for specifically naming "allergies" is that some people will consider allergies a medical issue and some will not, so the questioner wants to make sure the latter group answers about allergies anyway.)

Your second example uses any kind of, which is a different kind of phrase which emphasizes the totality of seasonings you are speaking of. Saying John does not use any kind of seasoning is stronger than saying John does not use seasoning. So it is not really an example of the general use of any.